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Unformatted text preview: Disjoint versus Independent events - Bayes rule Periklis A. Papakonstantinou ? University of Toronto Disjoint and independent events are two very different notions. We go through each and then we derive Bayes theorem. One thing that can go wrong when thinking about probability theory is to start thinking about probability theory in purely intuitive terms. The best thing to do when you first learn about probability theory is to disassociate in your mind any intuitive interpretation from what formally happens. As we will the theory is developed in such a way where in many cases the mathematical probabilities have some connection with interpretation of probabilities in real life. Here we wont discuss further this issue since it is a matter of the Philosophy of Mathematics (you may wish to check related courses in the Philosophy department). Events What is an event? Before starting discussing any type of probabilistic event we first fix the probability space we are talking. Anything else doesnt make any sense. The reason is that events are just subsets of the sample space. Nothing more and nothing less. Keep in mind that sometimes the sample space is implicitly defined through a precisely described exper- iment. Still at least in the beginning you are strongly encouraged to explicitly determine it. What is probability? A probability is a function given in the definition of the sample space. Though its not any function. In ECE190 we will only be concerned with (i) Discrete probability spaces. Let S be the sample space. Then, p : S [0 , 1] is a probability (mass) function if s S p ( s ) = 1. The elements of S are called elementary events . Let A be an event; i.e. A S . Then we define P ( A ) = a A p ( a ). In ECE190 we restrict further by considering also (ii) Uniform probability spaces. This makes life much simpler since the definition of P ( ) boils down to the following: Let A S be an event. Then, P ( A ) = | A | | S | . Example 1. Consider the following experiment. We roll an unbiased die. We wish to deter- mine the probability that the outcome is either the face 1 or the face 2. A probability space naturally associated with this experiment is S = { 1 , 2 , 3 , 4 , 5 , 6 } . The event of interest is A = { 1 , 2 } . Then, P ( A ) = 2 6 = 1 3 . ? This document possibly contains typos. Please submit any typos, remarks, suggestions to papakons@cs.toronto.edu . 2 Periklis Papakonstantinou (ECE 190, Fall 2006) Disjoint events Let S be a sample space. Two events A , B are disjoint if A B = . For example, in the previous example the event of having as an outcome 1 or 2 and the event of having as an outcome 3 are disjoint; since A = { 1 ,...
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